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''holomorphically-complete manifold''

A paracompact complex [[Analytic manifold|analytic manifold]]  $  M $
with the following properties:

1) for any compact set  $  K \subset  M $
the set

$$ 
\left \{ {
x \in X } : {| f( x) | \leq  \sup _ {z \in K }  | f( z) |  ( f \in {\mathcal O} ( M)) } \right \}
,
$$

where  $  {\mathcal O} ( M) $
is the algebra of holomorphic functions on  $  M $, 
is compact (holomorphic convexity);

2) for any two different points  $  x, y \in M $
there is a function  $  f \in {\mathcal O} ( M) $
such that  $  f( x) \neq f( y) $ (holomorphic separability);

3) in a neighbourhood of any point there is a holomorphic chart whose coordinate functions belong to  $  {\mathcal O} ( M) $.

The requirement of holomorphic convexity can be replaced by the following one: For any sequence  $  \{ {x _ {n} } : {n = 1, 2 , .  .  . } \} \subset  M $
without limit points there is a function  $  f \in {\mathcal O} ( M) $
such that  $  \sup _ {n}  | f( x _ {n} ) | = \infty $.

The class of Stein manifolds was introduced by K. Stein [[#References|[1]]] as a natural generalization of the notion of a [[Domain of holomorphy|domain of holomorphy]] in  $  \mathbf C  ^ {n} $. 
Any closed analytic submanifold in  $  \mathbf C  ^ {n} $
is a Stein manifold; conversely, any  $  n $-dimensional Stein manifold has a proper holomorphic imbedding in  $  \mathbf C  ^ {2n} $ (cf. [[Proper morphism|Proper morphism]]). Any non-compact Riemann surface is a Stein manifold. The direct generalization of a Stein manifold is a [[Stein space|Stein space]].

See also the references to [[Stein space|Stein space]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Stein,   "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem"  ''Math. Ann.'' , '''123'''  (1951)  pp. 201–222</TD></TR></table>